Gluons, quarks, letters, and envelopes

Yesterday I wrote a couple of posts about a combinatorics question that lead to OEIS sequence A000255. That page has this intriguing line:

This comment derives from a family of recurrences found by Malin Sjodahl for a combinatorial problem for certain quark and gluon diagrams (Feb 27 2010)

I love how pulling on a thread can lead you to unexpected places. What in the world does my little permutation problem have to do with particle physics

I found Malin Sjodahl's website, and apparently the citation above refers to this paper from 2009. The author's site doesn't list any papers from 2010. Maybe the paper was published electronically in 2009 and in print in 2010.

Counting tensors

The paper is mostly opaque to me since I don't know particle physics, but at one point Sjodahl says

The problem of finding all such topologies is equivalent to the number of ways of mapping N elements to each other without mapping a single one to itself ...

and says that the solution is

Sjodahl is not counting physical permutations but the number possible tensors associated with gluons and quark interaction diagrams.

The right-hand side above is essentially the same as the asymptotic estimate for the function Q(n) in the previous post.

I didn't find the recurrence that the OEIS comment alluded to. Perhaps I'm not looking at the same paper. Perhaps I'm not looking hard enough because I'm skimming a paper whose contents I don't understand.

The Montmort letter problem

In more mathematical terminology, Sjodahl is counting the number of permutations of N objects with no fixed point, known as the number derangements of a set N objects.

If you divide by the number of possible permutations N! you have the probability that a random permutation moves every object.

This was historically known as the Montmort letter problem, named after Pierre-Remond Montmort who asked the following question. IfN letters are randomly assigned to N envelopes, what is the probability that no letter ends up in the correct envelope?

The probability converges to 1/e as N approaches infinity. It approaches 1/e quickly, and so this is a good approximation even for moderate N. More on the rate of convergence in the next post.

Previous posts in this series

• Permutations with no consecutive elements
• Asymptotic behavior via generating functions

The post Gluons, quarks, letters, and envelopes first appeared on John D. Cook.